Definition of a convex cone

Question

In the definition of a convex cone, given that $x,y$ belong to the convex cone $C$,then $\theta_1x+\theta_2y$ must also belong to $C$, where $\theta_1,\theta_2 > 0$. What I don't understand is why there isn't the additional constraint that $\theta_1+\theta_2=1$ to make sure the line that crosses both $x$ and $y$ is restricted to the segment in between them.

Answer 1

Given that $x, y$ belong to the convex cone $C$, then $θ_1x+θ_2y$ must also belong to C, where $θ_1,θ_2 \geq 0 $.

Some visualization helped me to understand the equation. As you can see below, a point z within the convex cone can be represented as a linear combination of x and y with non-negative coefficients $θ_1$ and $θ_2$.

To be able to reach all points (i.e. write them as linear combinations of x and y with with non-negative coefficients $θ_1$ and $θ_2$) within the boundaries of the convex cone, we need to vary $θ_1$ and $θ_2$ without the constraint that they sum to 1.